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Abstract:

In this article we extend a previous definition of Castelnuovo–Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal B, obtaining, for each B, a B-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications. Even in the more restrictive framework where Castelnuovo–Mumford was defined before us, there were only very partial results on estimates for the shifts in a minimal graded free resolution from the Castelnuovo–Mumford regularity. We prove sharp estimates in our general framework, and this is one of our main advances. We provide tools to deduce information on the graded Betti numbers from the knowledge of regions where the local cohomology with support in a given graded ideal vanishes. Conversely, vanishing of local cohomology with support in any graded ideal is deduced from the shifts in a free resolution and the local cohomology of the polynomial ring. The flexibility of treating local cohomology with respect to any B opens up new possibilities for passing information. We provide new persistence results for the vanishing of local cohomology that extend the fact that weakly regular implies regular in the classical case, and we give sharp estimates for the regularity of a truncation of a module. In the last part, we present a result on Hilbert functions for multigraded polynomial rings, which provides a simple proof of the generalized Grothendieck–Serre formula. © 2016 Elsevier Inc.

Registro:

Documento: Artículo
Título:Castelnuovo Mumford regularity with respect to multigraded ideals
Autor:Botbol, N.; Chardin, M.
Filiación:Departamento de Matemática, FCEN, Universidad de Buenos Aires, Argentina
Institut de Mathématiques de Jussieu, UPMC, Paris Cedex 05, Boite 247, 4, place JussieuF-75252, France
Palabras clave:Castelnuovo–Mumford regularity; Local cohomology; Syzygies
Año:2017
Volumen:474
Página de inicio:361
Página de fin:392
DOI: http://dx.doi.org/10.1016/j.jalgebra.2016.11.017
Título revista:Journal of Algebra
Título revista abreviado:J. Algebra
ISSN:00218693
CODEN:JALGA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v474_n_p361_Botbol

Referencias:

  • Bagheri, A., Chardin, M., Tai Ha, H., The eventual shape of Betti tables of powers of ideals (2013) Math. Res. Lett., 20 (6), pp. 1033-1046
  • Bayer, D., Mumford, D., What can be computed in algebraic geometry? (1993) Computational Algebraic Geometry and Commutative Algebra, Cortona, 1991, Sympos. Math., XXXIV, pp. 1-48. , Cambridge Univ. Press Cambridge
  • Botbol, N., Implicitization of Rational Maps (2010), PhD Thesis Universidad de Buenos Aires & UPMC pages 1–169; Botbol, N., Implicit equation of multigraded hypersurfaces (2011) J. Algebra, 348 (1), pp. 381-401
  • Chardin, M., Some results and questions on Castelnuovo–Mumford regularity (2007) Syzygies and Hilbert Functions, Lect. Notes Pure Appl. Math., 254, pp. 1-40. , Chapman & Hall/CRC Boca Raton, FL
  • Chardin, M., Jouanolou, J.-P., Rahimi, A., The eventual stability of depth, associated primes and cohomology of a graded module (2013) J. Commut. Algebra, 5 (1), pp. 63-92
  • Colomé-Nin, G., Multigraded Structures and the Depth of Blow-up Algebras (2008), PhD Thesis Universitat de Barcelona; Cox, D.A., The homogeneous coordinate ring of a toric variety (1995) J. Algebraic Geom., 4 (1), pp. 17-50
  • Eisenbud, D., Goto, S., Linear free resolutions and minimal multiplicity (1984) J. Algebra, 88 (1), pp. 89-133
  • Hà, H.T., Multigraded regularity, a⁎-invariant and the minimal free resolution (2007) J. Algebra, 310 (1), pp. 156-179
  • Hà, H.T., Strunk, B., Minimal free resolutions and asymptotic behavior of multigraded regularity (2007) J. Algebra, 311 (2), pp. 492-510
  • Hartshorne, R., Local Cohomology (1967), a seminar given by A. Grothendieck, Harvard University, Fall, 1961 Springer-Verlag Berlin; Hilbert, D., Ueber die Theorie der algebraischen Formen (1890) Math. Ann., 36 (4), pp. 473-534
  • Hoffman, J.W., Wang, H.H., Castelnuovo–Mumford regularity in biprojective spaces (2004) Adv. Geom., 4 (4), pp. 513-536
  • Jayanthan, A.V., Verma, J.K., Grothendieck–Serre formula and bigraded Cohen–Macaulay Rees algebras (2002) J. Algebra, 254 (1), pp. 1-20
  • Maclagan, D., Smith, G.G., Multigraded Castelnuovo–Mumford regularity (2004) J. Reine Angew. Math., 571, pp. 179-212
  • Maclagan, D., Smith, G.G., Uniform bounds on multigraded regularity (2005) J. Algebraic Geom., 14 (1), pp. 137-164
  • Mumford, D., Lectures on Curves on an Algebraic Surface (1966) Ann. of Math. Stud., 59. , with a section by G.M. Bergman Princeton University Press Princeton, NJ
  • Mustaţǎ, M., Local cohomology at monomial ideals (2000) Symbolic Computation in Algebra, Analysis, and Geometry, J. Symbolic Comput., 29 (4-5), pp. 709-720. , Berkeley, CA, 1998
  • Mustaţă, M., Vanishing theorems on toric varieties (2002) Tohoku Math. J. (2), 54 (3), pp. 451-470
  • Roberts, P.C., Multiplicities and Chern Classes in Local Algebra (1998) Cambridge Tracts in Math., 133. , Cambridge University Press Cambridge
  • Sidman, J., Van Tuyl, A., Wang, H.H., Multigraded regularity: coarsenings and resolutions (2006) J. Algebra, 301 (2), pp. 703-727
  • Sidman, J., Van Tuyl, A., Multigraded regularity: syzygies and fat points (2006) Beitr. Algebra Geom., 47 (1), pp. 67-87
  • Sturmfels, B., On the Newton polytope of the resultant (1994) J. Algebraic Combin., 3 (2), pp. 207-236

Citas:

---------- APA ----------
Botbol, N. & Chardin, M. (2017) . Castelnuovo Mumford regularity with respect to multigraded ideals. Journal of Algebra, 474, 361-392.
http://dx.doi.org/10.1016/j.jalgebra.2016.11.017
---------- CHICAGO ----------
Botbol, N., Chardin, M. "Castelnuovo Mumford regularity with respect to multigraded ideals" . Journal of Algebra 474 (2017) : 361-392.
http://dx.doi.org/10.1016/j.jalgebra.2016.11.017
---------- MLA ----------
Botbol, N., Chardin, M. "Castelnuovo Mumford regularity with respect to multigraded ideals" . Journal of Algebra, vol. 474, 2017, pp. 361-392.
http://dx.doi.org/10.1016/j.jalgebra.2016.11.017
---------- VANCOUVER ----------
Botbol, N., Chardin, M. Castelnuovo Mumford regularity with respect to multigraded ideals. J. Algebra. 2017;474:361-392.
http://dx.doi.org/10.1016/j.jalgebra.2016.11.017